Curie Temperature of Gadolinium Experiment PDF

Preview

Summary

An experiment for advanced laboratory physics 407...

Description

(1/25/22) Ferromagnetism–The Curie Temperature of Gadolinium

Advanced Laboratory, Physics 407 University of Wisconsin Madison, Wisconsin 53706

Abstract The Curie temperature of Gadolinium is determined by measuring the magnetic susceptibility of a Gadolinium sample as a function of temperature. The data are analyzed using the Curie-Weiss law which contains the Curie temperature as a parameter. Gadolinium is unusual in that the Curie temperature is very close to room temperature.

1

1

Introduction

If a particular ferromagnetic material is heated above a characteristic temperature, now called the Curie temperature, it becomes paramagnetic.[1] The following table gives Curie temperatures for some ferromagnetic materials.

The distinctive property of ferromagnets is non-zero spontaneous magnetization: even when placed in zero external magnetic field, they remain “magnetic”. The disappearance of spontaneous magnetization in a ferromagnetic substance heated to its Curie temperature was first carefully measured by Weiss and Forrer in 1926.[2] The plot below of the data of Weiss and Forrer is from Statistical and Thermal Physics by F. Reif [3], Sec. 10.7.

2

Above the Curie temperature ferromagnetic materials are paramagnetic: when placed in an external magnetic field, they develop a magnetization M parallel to the field. The classical theory of paramagnetism treats the substance as a collection of magnetic dipoles with no interactions between them. In an external magnetic field, each magnetic dipole has a potential energy given by: E = −µ · H = −µH cos θ

(1)

where µ is the magnetic moment of the dipole (magnitude µ), H is the applied magnetic field and θ is the angle between the dipole and the direction of H. If there are N dipoles per unit volume the magnetization would be given by M = N µ where the direction of the magnetization would be that of the applied field. However in the presence of thermal agitation it is necessary to use the Boltzmann distribution to average over the dipole distribution in thermal equilibrium at temperature T. We then have: M = N µcos θ = N µ

Z

e−E/kT cos θdΩ/

Z

e−E/kT dΩ

(2)

where dΩ is the element of solid angle and e−E/kT is the Boltzmann distibution of a dipole at angle θ with respect to the applied field at absolute temperature T. The integration yields a result in terms of the Langevin function L(x) = coth x − 1/x with x = µH/kT giving: M = N µL(x).

(3)

When x ≪ 1 the Langevin function L(x) approaches x/3 so that Eq. 3 becomes: (4) M∼ = N µ2 H/3kT . This is a very good approximation except at low temperature or extremely high magnetic fields and gives us the Curie law for paramagnetism χ ≡ M/H = C/T [1]. The classical theory of paramagnetism was extended to cover ferromagnetism by Weiss in 1907 [4] . The basis of the Weiss molecular theory of ferromagnetism is that below the Curie temperature, a ferromagnet is composed of small, spontaneously magnetized regions called domains and the total magnetic moment of the material is the vector sum of the magnetic 3

moments of the individual domains. Each domain is magnetized due to the strong magnetic interaction within the domain which tends to align the individual magnetic moments within the domain. The spontaneous magnetization below the Curie temperature comes about from an internal magnetic field called the Weiss molecular field which is proportional to the magnetization of the domain. Thus the effective field acting on any magnetic moment within the domain may be written as: H = H0 + λM

(5)

where H0 is an externally applied field and λM is the Weiss molecular field whose order of magnitude in iron is the surprisingly large 107 oersteds. Above the Curie temperature, the Curie law now becomes: M/(H0 + λM ) = C/T .

(6)

Solving for χ = M/H0 we obtain: χ = M/H0 = C/(T − Cλ) = C/(T − TC )

(7)

which defines the Curie temperature as TC = Cλ giving us the Curie-Weiss law for the paramagnetic behavior of ferromagnetic substances above the Curie temperature. We see that the Curie-Weiss law leads to a nonzero magnetization when H0 = 0 at T = TC . Our contemporary understanding of ferromagnetism is based on the quantum mechanical theory developed by Heisenberg in 1928 [5]. The Heisenberg theory is based on an effective interaction between the electron spins, the exchange interaction energy, given by: Exchange Energy = −2Jij Si · Sj

(8)

where Si and Sj are the spin angular momentum vectors of two electrons i and j, and Jij is the exchange integral between the electrons. Depending on the balance between the Pauli Principle and the electrostatic interaction energy of the electrons, the exchange integral may be positive or negative corresponding to parallel spins (ferromagnetism) or antiparallel spins (antiferromagnetism or ferrimagnetism). The order of magnitude of J is kTC ∼ 0.025 eV for Gadolinium. Further reading: A fully quantum-mechanical treatment of paramagnetism and ferromagnetism is given in chapters 14 and 15 of reference [7]. 4

2

Measurement

As seen above, the magnetic susceptibility of a ferromagnet above its Curie temperature is given by the Curie-Weiss law: C (9) T − TC where χ and µ are the magnetic susceptibility and relative magnetic permeability of the material respectively. C is a constant characteristic for a given substance and TC is the Curie temperature. Eq. 9 is only valid above the Curie temperature. The relative magnetic susceptibility of a material is readily determined by placing a sample of the material inside a small coil and measuring the inductance of the coil with and without the sample. If the inductance is measured as a function of temperature from above to below the Curie temperature, the Curie-Weiss law, Eq. 9, can be used to determine the Curie temperature. [6] The relative magnetic permeability of the sample can be written as χ= µ−1=

µ=

L(T ) L0

(10)

where L(T ) is the inductance of the coil at temperature T and L0 is the inductance of the coil without the sample. This is not exactly the relative permeability since not all the magnetic flux will couple to the sample. From Eq. 9 we can write an equation linear in T as !−1

L(T ) −1 L0

=

T − TC . C

(11)

The right hand side of this equation is zero when T = TC so a plot of the left hand side vs temperature extrapolated to zero will intersect the x-axis at T = TC . The lack of a 100% fill factor for the coil will not affect this result.

3 3.1

Apparatus Coil

The coil is from a commercial relay and has an inductance of about 2.16 mH. 5

3.2

Gadolinium Sample

The Gadolinium sample is in the form of a rod 6.35 mm in diameter and length 10 mm. The Gadolinium is 99.9% pure.

3.3

Thermocouple Thermometer

The thermometer is a Fluke Model 51 type K thermocouple thermometer. with a range of −200◦ C to 1370◦ C. The rated accuracy is ±(0.1% of reading + 0.7◦ C).

3.4

Thermoelectric Cooler

The Thermoelectric Cooler is an Advanced Thermoelectric model TCP-30. This device is a solid-state heat pump based on the Peltier Effect: a DC current applied to a thermoelectric module composed of two dissimilar materials (p-type and n-type bismuth-telluride semiconductors) causes a temperature differential. It looks like a hot plate but can go up or down in temperature: if the current flows one way, the plate heats up, if it flows the other way the plate cools down. The TCP-30 has a control temperature range of −20◦ C to 100◦ C, the actual operational temperature range being set by the heat load, and is controlled by a separate temperature controller which supplies the power to the TCP-30. The maximum TCP-30 power rating is 12V at 6A.

3.5

Programmable Temperature Controller

The Temperature Controller is an Oven Industries model 5C7-362. The controller is computer controlled and supplies power to the Thermoelectric Cooler from a 12 V DC power supply. The controller output signal to the cooler is Pulse Width Modulated at 675 Hz and sets the average power output to maintain a set-point temperature that has been entered into the control software. The sample temperature is measured by a thermistor mounted inside the coil together with the sample. The thermistor signal is fed back to the temperature controller and the software compares the thermistor temperature to the set-point temperature, signaling the controller to supply just enough power to make the thermistor and set-point temperatures equal. This control scheme can control the temperature to ±0.1◦ C at the control sensor. 6

3.6

Thermistor Sensor

The thermistor is an Oven Industries model TS67 with a base resistance of 15,000 ohms at 25◦ C. The temperature accuracy as determined from the look-up table in the computer program is ±1.0◦ C over a 0◦ C to 100◦ C temperature range. However the temperature scale can be independently calibrated and the result programmed into the controller. The reference temperature is taken to be 25◦ C since the thermistor input for the reference temperature is replaced by a 15k resistor. Since the temperature of this resistor is not necessarily at 25◦ C, an INPUT 1 OFFSET temperature must be entered.

3.7

Control Circuit

The power supply and temperature controller are mounted in a control circuit box which has an ON-OFF switch and an ammeter that is zero centered and reads the current to the temperature controller. A negative current cools and a positive current heats the Cooler. This current starts out high and gradually drops to zero as the temperature set-point is reached.

3.8

Inductance Meter

The inductance meter is a Atlas model LCR45 Impedance Meter. It measures the impedance of the component to which it is connected by subjecting it to an AC signal (|V | ≤ 1.05 V, |I| ≤ 3.25 mA). For our inductor the AC frequency should be set to 14.9254 kHz.

4

Experiment 1. Calibrate the thermocouple thermometer in an ice-water equilibrium mixture. 2. Calibrate the Temperature Controller by placing the thermistor and the thermocouple thermometer in the oil bath at room temperature. You will have to read the temperature reported by the thermistor using the software described below. If the thermometer and thermistor readings do not agree, an offset temperature parameter can be entered into the software. 7

3. Measure L0 , the inductance of the coil without the Gadolinium sample. 4. Carefully place the Gadolinium sample and the thermistor inside the coil. The whole assembly is placed in a small beaker filled with mineral oil which serves as a heat sink. Place the beaker on the cooler plate. 5. Open the controller program MC362.exe located as an icon on the desktop. Read the documentation so you are sure you know what the following buttons do: “SEND BOX VALUES”, “CONTROL TYPE”, and “FIXED SET TEMP.” In particular note that if “CONTROL TYPE” is set to “COMPUTER CONTROL” then one enters a dimensionless number in the “FIXED SET TEMP” box, where +12 corresponds to full heating current (about 3.7 A) and −12 corresponds to full cooling current (about −3.7 A). Note there are several “SEND BOX VALUES” buttons. Which box displays the temperature measured by the thermistor? 6. Enter values in boxes as per the screenshot below.

7. The largest source of uncertainty in this lab is due to the temperature gradient across the Gadolinium sample. To get accurate data you need to keep this temperature gradient as small as possible (figure of merit: less than 2◦ C difference between all temperatures). 8. One strategy to obtain good results is to heat the sample a few degrees hotter than you need for your measurements, and then let it equilibrate 8

toward ambient temperature while taking readings of the inductance (and the inverse for measurements below ambient temperature). Stirring the oil helps reduce temperature gradients. Avoid dislodging the thermistor. 9. Measure the inductance of the coil as a function of temperature in steps of 0.1◦ C from 17◦ to 27◦ C. Try to keep the difference the thermistor and thermocouple temperatures less than 2◦ C. Record all temperatures. 10. Eq. 11 is a linear relation (y vs x) between measured quantities and temperature with the x-intercept giving T = TC when y = 0. Plot the left hand side vs temperature and the result will show a linear T − TC behavior above the Curie temperature. Extrapolate the linear portion of the plot to y = 0 and determine the temperature intercept. You will have to use some judgement as to how much data to include in the linear extrapolation. Remember that your result for the Curie temperature is no better than your temperature calibration of the sample temperature. A computer least squares fit of the data is appropriate. Your result should include your result for the Curie temperature with uncertainty and a value for the chi-square of the fit.

References [1] P. Curie, “Propri´et´es magn´etiques des corps a` diverses temp´eratures,” Ann. Chim. Phy. 7, 289-404 (1895). [2] P. Weiss and R. Forrer, “Aimantation et Ph´enom`ene Magn´etocalorique du Nickel,” Ann. Phys., 10, 153-213 (1926). [3] F. Reif. Fundamentals of Statistical and Thermal Physics, Waveland Press, 2008. [4] P. Weiss, “L’hypoth`ese du champ mol´eculaire et la propri´et´e ferromagn´etique” J. Phys. 6, 661-690 (1907). [5] W. Heisenberg, “Zur Theorie des Ferromagnetismus,” Zeitschrift f˝ ur Physik 49, 619-636 (1928).

9

[6] T Lewowski and K Wo´zniak, “Measurement of Curie temperature for Gadolinium: a laboratory experiment for students,” Eur. J. Phys. 18 453-455 (1997). [7] C. Kittel. Introduction to Solid State Physics 7th ed., John Wiley & Sons, Inc., 1996.

10...